Optimal. Leaf size=59 \[ \frac{2^{2 n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}} \]
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Rubi [A] time = 0.0156874, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2651} \[ \frac{2^{2 n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2651
Rubi steps
\begin{align*} \int (2-2 \sin (c+d x))^n \, dx &=\frac{2^{\frac{1}{2}+2 n} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1+\sin (c+d x))\right )}{d \sqrt{1-\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.106498, size = 90, normalized size = 1.53 \[ \frac{\cos (c+d x) (2-2 \sin (c+d x))^n \cos ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )^{-n-\frac{1}{2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left (\frac{1}{4} (2 c+2 d x-\pi )\right )\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.212, size = 0, normalized size = 0. \begin{align*} \int \left ( 2-2\,\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-2 \, \sin \left (d x + c\right ) + 2\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-2 \, \sin \left (d x + c\right ) + 2\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 - 2 \sin{\left (c + d x \right )}\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-2 \, \sin \left (d x + c\right ) + 2\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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